Note
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Download Jupyter notebook:
https://docs.doubleml.org/stable/examples/py_double_ml_cate.ipynb.
Python: Conditional Average Treatment Effects (CATEs)#
In this simple example, we illustrate how the DoubleML package can be used to estimate conditional average treatment effects with B-splines for one or two-dimensional effects.
Data#
We define a data generating process to create synthetic data to compare the estimates to the true effect. The data generating process is based on the Monte Carlo simulation from this paper and this implementation from EconML.
[1]:
import numpy as np
import pandas as pd
import doubleml as dml
The data is generated as
$ \begin{align} Y_i & = g(X_i)T_i + \langle W_i,\gamma_0\rangle + \epsilon_i \\ T_i & = \langle W_i,\beta_0\rangle +\eta_i, \end{align} $
where \(W_i\sim\mathcal{N}(0,I_{d_w})\), \(X_i\sim\mathcal{U}[0,1]^{d_x}\) and \(\epsilon_i,\eta_i\sim\mathcal{U}[0,1]\). The coefficient vectors \(\gamma_0\) and \(\beta_0\) both have small random support which values are drawn independently from \(\mathcal{U}[0,1]\). Further, \(g(x)\) defines the conditional treatment effect, which is defined differently depending on the dimension of \(x\).
If \(x\) is univariate the conditional treatment effect takes the following form
whereas for a two-dimensional variable \(x=(x_1,x_2)\) the conditional treatment effect is defined as
[2]:
def treatment_effect_1d(x):
te = np.exp(2 * x) + 3 * np.sin(4 * x)
return te
def treatment_effect_2d(x):
te = np.exp(2 * x[0]) + 3 * np.sin(4 * x[1])
return te
def create_synthetic_data(n_samples=200, n_w=30, support_size=5, n_x=1):
"""
Creates a simple synthetic example for conditional treatment effects.
Parameters
----------
n_samples : int
Number of samples.
Default is ``200``.
n_w : int
Dimension of covariates.
Default is ``30``.
support_size : int
Number of relevant covariates.
Default is ``5``.
n_x : int
Dimension of treatment variable.
Default is ``1``.
Returns
-------
data : pd.DataFrame
A data frame.
"""
# Outcome support
# With the next two lines we are effectively choosing the matrix gamma in the example
support_y = np.random.choice(np.arange(n_w), size=support_size, replace=False)
coefs_y = np.random.uniform(0, 1, size=support_size)
# Define the function to generate the noise
epsilon_sample = lambda n: np.random.uniform(-1, 1, size=n_samples)
# Treatment support
# Assuming the matrices gamma and beta have the same non-zero components
support_t = support_y
coefs_t = np.random.uniform(0, 1, size=support_size)
# Define the function to generate the noise
eta_sample = lambda n: np.random.uniform(-1, 1, size=n_samples)
# Generate controls, covariates, treatments and outcomes
w = np.random.normal(0, 1, size=(n_samples, n_w))
x = np.random.uniform(0, 1, size=(n_samples, n_x))
# Heterogeneous treatment effects
if n_x == 1:
te = np.array([treatment_effect_1d(x_i) for x_i in x]).reshape(-1)
elif n_x == 2:
te = np.array([treatment_effect_2d(x_i) for x_i in x]).reshape(-1)
# Define treatment
log_odds = np.dot(w[:, support_t], coefs_t) + eta_sample(n_samples)
t_sigmoid = 1 / (1 + np.exp(-log_odds))
t = np.array([np.random.binomial(1, p) for p in t_sigmoid])
# Define the outcome
y = te * t + np.dot(w[:, support_y], coefs_y) + epsilon_sample(n_samples)
# Now we build the dataset
y_df = pd.DataFrame({'y': y})
if n_x == 1:
x_df = pd.DataFrame({'x': x.reshape(-1)})
elif n_x == 2:
x_df = pd.DataFrame({'x_0': x[:,0],
'x_1': x[:,1]})
t_df = pd.DataFrame({'t': t})
w_df = pd.DataFrame(data=w, index=np.arange(w.shape[0]), columns=[f'w_{i}' for i in range(w.shape[1])])
data = pd.concat([y_df, x_df, t_df, w_df], axis=1)
covariates = list(w_df.columns.values) + list(x_df.columns.values)
return data, covariates, te
One-dimensional Example#
We start with \(X\) being one-dimensional and create our training data.
[3]:
# DGP constants
np.random.seed(42)
n_samples = 2000
n_w = 10
support_size = 5
n_x = 1
# Create data
data, covariates, true_effect = create_synthetic_data(n_samples=n_samples, n_w=n_w, support_size=support_size, n_x=n_x)
data_dml_base = dml.DoubleMLData(data,
y_col='y',
d_cols='t',
x_cols=covariates)
Next, define the learners for the nuisance functions and fit the IRM Model. Remark that the learners are not optimal for the linear form of this example.
[4]:
# First stage estimation
from sklearn.ensemble import RandomForestClassifier, RandomForestRegressor
randomForest_reg = RandomForestRegressor(n_estimators=500)
randomForest_class = RandomForestClassifier(n_estimators=500)
np.random.seed(42)
dml_irm = dml.DoubleMLIRM(data_dml_base,
ml_g=randomForest_reg,
ml_m=randomForest_class,
trimming_threshold=0.01,
n_folds=5)
print("Training IRM Model")
dml_irm.fit()
Training IRM Model
[4]:
<doubleml.double_ml_irm.DoubleMLIRM at 0x7f9918c5a820>
To estimate the CATE, we rely on the best-linear-predictor of the linear score as in Semenova et al. To approximate the target function \(g(x)\) with a linear form, we have to define a data frame of basis functions. Here, we rely on patsy to construct a suitable basis of B-splines.
[5]:
import patsy
design_matrix = patsy.dmatrix("bs(x, df=5, degree=2)", {"x":data["x"]})
spline_basis = pd.DataFrame(design_matrix)
To estimate the parameters to calculate the CATE estimate call the cate() method and supply the dataframe of basis elements.
[6]:
cate = dml_irm.cate(spline_basis)
print(cate)
================== DoubleMLBLP Object ==================
------------------ Fit summary ------------------
coef std err t P>|t| [0.025 0.975]
0 0.800985 0.187226 4.278164 1.973568e-05 0.433805 1.168165
1 2.317576 0.312855 7.407836 1.885192e-13 1.704020 2.931132
2 4.731110 0.199998 23.655730 2.996077e-109 4.338882 5.123338
3 4.501506 0.239412 18.802350 9.746797e-73 4.031982 4.971030
4 3.865116 0.245950 15.715036 1.485625e-52 3.382770 4.347463
5 4.114845 0.266575 15.435994 7.270658e-51 3.592051 4.637639
To obtain the confidence intervals for the CATE, we have to call the confint() method and a supply a dataframe of basis elements. This could be the same basis as for fitting the CATE model or a new basis to e.g. evaluate the CATE model on a grid. Here, we will evaluate the CATE on a grid from 0.1 to 0.9 to plot the final results. Further, we construct uniform confidence intervals by setting the option joint and providing a number of bootstrap repetitions n_rep_boot.
[7]:
new_data = {"x": np.linspace(0.1, 0.9, 100)}
spline_grid = pd.DataFrame(patsy.build_design_matrices([design_matrix.design_info], new_data)[0])
df_cate = cate.confint(spline_grid, level=0.95, joint=True, n_rep_boot=2000)
print(df_cate)
2.5 % effect 97.5 %
0 2.161123 2.486602 2.812081
1 2.279878 2.607058 2.934238
2 2.394024 2.725133 3.056242
3 2.504552 2.840828 3.177103
4 2.612308 2.954141 3.295974
.. ... ... ...
95 4.480207 4.811954 5.143701
96 4.483368 4.809714 5.136060
97 4.487456 4.808597 5.129738
98 4.491828 4.808603 5.125377
99 4.495728 4.809731 5.123734
[100 rows x 3 columns]
Finally, we can plot our results and compare them with the true effect.
[8]:
from matplotlib import pyplot as plt
plt.rcParams['figure.figsize'] = 10., 7.5
df_cate['x'] = new_data['x']
df_cate['true_effect'] = treatment_effect_1d(new_data['x'])
fig, ax = plt.subplots()
ax.plot(df_cate['x'],df_cate['effect'], label='Estimated Effect')
ax.plot(df_cate['x'],df_cate['true_effect'], color="green", label='True Effect')
ax.fill_between(df_cate['x'], df_cate['2.5 %'], df_cate['97.5 %'], color='b', alpha=.3, label='Confidence Interval')
plt.legend()
plt.title('CATE')
plt.xlabel('x')
_ = plt.ylabel('Effect and 95%-CI')
Two-Dimensional Example#
It is also possible to estimate multi-dimensional conditional effects. We will use the same data-generating process as above, but let \(X\) be two-dimensional.
[9]:
# DGP constants
np.random.seed(42)
n_samples = 5000
n_w = 10
support_size = 5
n_x = 2
[10]:
# Create data
data, covariates, true_effect = create_synthetic_data(n_samples=n_samples, n_w=n_w, support_size=support_size, n_x=n_x)
data_dml_base = dml.DoubleMLData(data,
y_col='y',
d_cols='t',
x_cols=covariates)
As univariate example estimate the IRM Model.
[11]:
# First stage estimation
from sklearn.ensemble import RandomForestClassifier, RandomForestRegressor
randomForest_reg = RandomForestRegressor(n_estimators=500)
randomForest_class = RandomForestClassifier(n_estimators=500)
np.random.seed(123)
dml_irm = dml.DoubleMLIRM(data_dml_base,
ml_g=randomForest_reg,
ml_m=randomForest_class,
trimming_threshold=0.01,
n_folds=5)
print("Training IRM Model")
dml_irm.fit()
Training IRM Model
[11]:
<doubleml.double_ml_irm.DoubleMLIRM at 0x7f9914008ac0>
As above, we will rely on the patsy package to construct the basis elements. In the two-dimensional case, we will construct a tensor product of B-splines (for more information see here).
[12]:
design_matrix = patsy.dmatrix("te(cr(x_0, df=6), cc(x_1, df=6)) - 1", {"x_0": data["x_0"], "x_1": data["x_1"]})
spline_basis = pd.DataFrame(design_matrix)
cate = dml_irm.cate(spline_basis)
print(cate)
================== DoubleMLBLP Object ==================
------------------ Fit summary ------------------
coef std err t P>|t| [0.025 0.975]
0 0.604233 0.332103 1.819412 6.890880e-02 -0.046836 1.255302
1 3.125605 0.307803 10.154575 5.417216e-24 2.522176 3.729034
2 4.451022 0.324940 13.697978 5.996182e-42 3.813996 5.088048
3 4.213811 0.313247 13.452019 1.525932e-40 3.599707 4.827914
4 2.314430 0.318764 7.260634 4.450491e-13 1.689512 2.939349
5 0.699531 0.287962 2.429252 1.516527e-02 0.134999 1.264063
6 0.803563 0.170947 4.700667 2.663304e-06 0.468432 1.138694
7 3.463561 0.171227 20.227921 1.715967e-87 3.127881 3.799241
8 4.282365 0.168945 25.347725 2.107199e-133 3.951159 4.613572
9 3.966890 0.179375 22.115112 1.933984e-103 3.615236 4.318543
10 2.608630 0.171217 15.235777 2.910942e-51 2.272968 2.944292
11 0.844913 0.170626 4.951832 7.595500e-07 0.510410 1.179416
12 1.198591 0.174801 6.856877 7.898403e-12 0.855903 1.541279
13 4.461987 0.167341 26.663976 1.666402e-146 4.133924 4.790050
14 4.917113 0.163506 30.072978 1.124828e-182 4.596569 5.237657
15 4.952606 0.166034 29.828898 5.533714e-180 4.627106 5.278106
16 3.522473 0.166132 21.202881 1.389062e-95 3.196781 3.848165
17 1.394159 0.170747 8.165059 4.037491e-16 1.059420 1.728899
18 2.573164 0.169213 15.206629 4.455693e-51 2.241431 2.904896
19 5.395435 0.171672 31.428778 6.708865e-198 5.058882 5.731987
20 6.420662 0.173736 36.956496 2.478373e-264 6.080063 6.761260
21 6.133043 0.165315 37.099054 3.949421e-266 5.808952 6.457134
22 4.625297 0.165425 27.960039 6.916526e-160 4.300991 4.949604
23 2.348626 0.162656 14.439233 2.516821e-46 2.029748 2.667503
24 4.176464 0.163197 25.591588 8.597218e-136 3.856526 4.496402
25 7.024126 0.161962 43.369002 0.000000e+00 6.706609 7.341643
26 7.713400 0.171418 44.997519 0.000000e+00 7.377345 8.049456
27 7.544198 0.166797 45.229898 0.000000e+00 7.217203 7.871193
28 5.975691 0.175875 33.976913 1.070637e-227 5.630898 6.320483
29 4.329680 0.158862 27.254263 1.531250e-152 4.018240 4.641121
30 7.108494 0.351517 20.222310 1.906151e-87 6.419364 7.797623
31 9.673541 0.344538 28.076812 4.087311e-161 8.998093 10.348988
32 10.648766 0.359216 29.644502 5.842619e-178 9.944544 11.352987
33 10.022048 0.312943 32.025133 9.732374e-205 9.408541 10.635555
34 9.230270 0.307361 30.030686 3.300708e-182 8.627706 9.832834
35 6.120285 0.319289 19.168511 4.588444e-79 5.494339 6.746232
Finally, we create a new grid to evaluate and plot the effects.
[13]:
grid_size = 100
x_0 = np.linspace(0.1, 0.9, grid_size)
x_1 = np.linspace(0.1, 0.9, grid_size)
x_0, x_1 = np.meshgrid(x_0, x_1)
new_data = {"x_0": x_0.ravel(), "x_1": x_1.ravel()}
[14]:
spline_grid = pd.DataFrame(patsy.build_design_matrices([design_matrix.design_info], new_data)[0])
df_cate = cate.confint(spline_grid, joint=True, n_rep_boot=2000)
print(df_cate)
2.5 % effect 97.5 %
0 1.642251 2.103445 2.564639
1 1.657373 2.116454 2.575535
2 1.667890 2.130064 2.592238
3 1.675004 2.144320 2.613636
4 1.679988 2.159267 2.638546
... ... ... ...
9995 4.180079 4.651675 5.123270
9996 4.261262 4.732416 5.203570
9997 4.337863 4.813105 5.288347
9998 4.409008 4.893745 5.378481
9999 4.474072 4.974339 5.474605
[10000 rows x 3 columns]
[15]:
import plotly.graph_objects as go
true_effect = np.array([treatment_effect_2d(x_i) for x_i in zip(x_0.ravel(), x_1.ravel())]).reshape(x_0.shape)
effect = np.asarray(df_cate['effect']).reshape(x_0.shape)
lower_bound = np.asarray(df_cate['2.5 %']).reshape(x_0.shape)
upper_bound = np.asarray(df_cate['97.5 %']).reshape(x_0.shape)
fig = go.Figure(data=[
go.Surface(x=x_0,
y=x_1,
z=true_effect),
go.Surface(x=x_0,
y=x_1,
z=upper_bound, showscale=False, opacity=0.4,colorscale='purp'),
go.Surface(x=x_0,
y=x_1,
z=lower_bound, showscale=False, opacity=0.4,colorscale='purp'),
])
fig.update_traces(contours_z=dict(show=True, usecolormap=True,
highlightcolor="limegreen", project_z=True))
fig.update_layout(scene = dict(
xaxis_title='X_0',
yaxis_title='X_1',
zaxis_title='Effect'),
width=700,
margin=dict(r=20, b=10, l=10, t=10))
fig.show()